Question

Assertion (A): The system of linear equations 3x – 5y + 7 = 0 and –6x + 10y + 14 = 0 is inconsistent.

Reason (R): When two linear equations don’t have unique solution, they always represent parallel lines.

विकल्प

• Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).

• Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).

• Assertion (A) is true, but Reason (R) is false.

• Assertion (A) is false, but Reason (R) is true.

Answer 1

Assertion (A) is true, but Reason (R) is false.

Explanation:

Evaluate Assertion (A):

A system of linear equations is inconsistent if it has no solution. This occurs when the lines are parallel.

For equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, the condition for no solution is:

`(a_1)/(a_2) = (b_1)/(b_2) ≠ (c_1)/(c_2)`

Evaluating Assertion (A):

Equation 1: 3x – 5y + 7 = 0

⇒ a₁ = 3, b₁ = –5, c₁ = 7

Equation 2: –6x + 10y + 14 = 0

⇒ a₂ = –6, b₂ = 10, c₂ = 14

Check the ratios:

`(a_1)/(a_2) = 3/(-6) = - 1/2`

`(b_1)/(b_2) = (-5)/10 = - 1/2`

`(c_1)/(c_2) = 7/14 = 1/2`

Since `(a_1)/(a_2) = (b_1)/(b_2) ≠ (c_1)/(c_2) (-1/2 = - 1/2 ≠ 1/2)`, the system has no solution.

Thus, the system is inconsistent. 

Assertion (A) is true.

Evaluating Reason (R):

A system “doesn’t have a unique solution” when it has either no solution (parallel lines) or infinitely many solutions (coincident lines).

Therefore, it is incorrect to say it always represents parallel lines.

Reason (R) is false.